What is a non-manifold vertex?
For anyone else with this problem the detailing of this is actually outlined in the blender docs here
The docs say:

Vertices:
Selects vertices that belong to wire and multiple face edges, isolated vertices, and vertices that belong to non-adjoining faces.

In the attempt not to repeat the good descriptions that exist elsewhere on this stack:

Wire edges: What are wire edges?
Multiple face edges:



Isolated vertices: https://en.wikipedia.org/wiki/Vertex_(graph_theory)#Types_of_vertices
Vertices which share non-adjoining faces:


N.B: if you hover over each option in the Select all by Trait menu you get a tooltip about each option:

This is all my best answer as to each scenario, if edits are required please go ahead